Formulae for order one invariants of immersions and embeddings of surfaces

The universal order 1 invariant f^U of immersions of a closed orientable surface into R^3, whose existence has been established in [N3], takes values in the group G_U = K \oplus Z/2 \oplus Z/2 where K is a countably generated free Abelian group. The projections of f^U to K and to the first and second Z/2 factors are denoted f^K, M, Q respectively. An explicit formula for the value of Q on any embedding has been given in [N2]. In the present work we give an explicit formula for the value of f^K on any immersion, and for the value of M on any embedding.